= Solution
The last two terms in the preceding result equal $v_s$. Therefore
$$
v_{tt}=y^2u_{xx}-2xyu_{xy}+x^2u_{yy}+v_s.
$$
The left-hand side of the original <partial differential equation> is consequently $v_{tt}-v_s$, giving
$$
\boxed{v_{tt}-v_s=0,\qquad\text{equivalently }v_s=v_{tt}.}
$$
In these variables it has the form of a <heat equation>, with logarithmic radial coordinate $s$ playing the evolution variable and angular coordinate $t$ playing the spatial variable. If $u$ is single-valued around the whole punctured <plane>, then $v(s,t+2\pi)=v(s,t)$ as well; that periodicity is a global coordinate condition, not an extra local transformation term.
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